We theoretically investigate mid-IR absorption enhancement with a SiC one-dimensional photonic crystal (PC) microstructure at the frequency regime of the phonon-polariton band gap, where efficient absorption is unattainable in the bulk material. Our study reveals an intricate relationship between absorption efficiency and the energy velocity of light propagation, that is far more complex than hitherto believed. In particular, our findings suggest that absorption peaks away from the photonic-crystal band edge where energy velocity is minimum. While efficient absorption is still associated with a slow-light mode, the latter is faster by at least an order of magnitude in comparison to the bulk material. Moreover, our calculations suggest that absorption becomes optimal when light gradually slow downs as it enters the PC. Relying on this insight, we achieved near-perfect absorption around the phonon-polariton mid-gap frequency with a PC with a suitably terminated end face. We further demonstrate that the near-perfect absorptive property can be tuned with the incident light angle, to be polarization insensitive or polarization selective. We believe our proposed non-metallic paradigm opens up a new route for harnessing infrared absorption with semiconductor and ionic-crystal materials.
© 2012 OSA
Harnessing absorption at visible frequencies with nanostructured materials has been a subject of numerous current research efforts [1–9] driven by the strong interest to enhance the performance of photovoltaic devices. The design principles employed by these works to optimize the photon absorption involve enhancement of the near-field by excitation of surface or waveguide electromagnetic (EM) modes [3, 5, 6, 8], engineering lossy material surfaces with ultra-low reflectivity [2, 4, 9], or directing the EM energy into slow waveguide modes that would imply a large interaction time between light and matter . However, manipulating photonic absorption at mid-IR frequencies is also an extremely desirable functionality pertinent to biosensing and medical diagnostic applications. Typical pathogens have their absorption fingerprint around 10 μm . Also, infrared imaging systems [11, 12] have been recently proposed for cancer diagnosis at early stages [11, 13]. Thus far, mainly metallic microstructures have been employed to achieve absorption enhancement at mid-IR frequencies. The pioneering proposal of S. Y. Lin  et al. reported an absorptance as high as 50% with a tungsten (W) three-dimensional layer-by-layer PC. The latter was attributed to a large photon-matter interaction time emanating from a slow Bloch mode , occurring at the PC band edge, and a long material path within the PC.
Metallic PCs are typically highly reflective at mid-IR frequencies, and accordingly have significant limitations with respect to absorption enhancement. In particular, at this frequency range bulk metals are near-perfect, with a skin depth of the order of λ0/300, with λ0 being the free space wavelength, and essentially a 100% reflectivity. Only very recently, J. A. Mason et al.  achieved a 100% absorption with an Au-SOG (spin-on glass)-Au periodic strip structure, and demonstrated its potential as a bio-sensing platform . The latter perfect absorbing system relied on the excitation of the composite waveguide modes, and is exploiting absorption in both the Au and SOG layers . Given the immense need for mid-IR absorption manipulation, it is of utmost interest to explore alternate schemes that can yield a near-perfect absorption in this frequency range.
Here we report for the first time a non-metallic route to near-perfect mid-IR absorption by employing a semiconductor-based PC. We will analyze carefully the design principles that lead to such a near-perfect absorption in a one-dimensional PC structure. Semiconductor materials have their phonon-polariton gap in the infrared spectrum  and have a high loss-tangent within this frequency region. The bulk structures are still highly reflective, but considerably better impedance matched with vacuum than metals are. Also their respective skin depth in the mid-IR spectrum is at least an order of magnitude higher, which means they can offer a larger effective light-matter interaction path in comparison to their metallic counterparts.
In particular, we present the periodic SiC PC system under study in Sec. 2 and briefly discuss the theoretical method employed in our calculations. We present the results obtained for the absorptance in Sec. 3, and discuss their correlation to intensity enhancement in the SiC layers. In Sec. 4 we analyze the intricate role of the energy velocity of the in-coupled electromagnetic (EM) wave in association to absorptance. Based on the insight gained by this analysis, we propose in Sec. 5 a suitable PC with a terminated end face that exhibits a near-perfect mid-IR absorption. This result essentially represents a dramatic absorption enhancement, about 30 times larger in comparison to the bulk SiC structure. We further explore the capability of this paradigm to demonstrate polarization insensitive as well as polarization selective near-perfect absorption in Sec. 6. Finally, we discuss our conclusions in Sec. 7.
2. The one-dimensional SiC photonic crystal system modeled with TMM
We choose SiC (Silicon Carbide) for the semiconductor constituent, because its phonon-polariton mid-gap is around 11μm, where the absorption fingerprint of many biological substances lies [10, 19]. The periodic one-dimensional (1D) PC comprises of alternating SiC and air slabs, of widths d1 and d2 respectively and has a unit cell size a = d1 + d2. The ratio f = d1/a represents the SiC filling ratio . In all of the following, we consider a structural unit cell with a=10.3μm. In Fig. 1(a), the first two unit cells of the structure are depicted. As seen in the figure, we consider normal incidence along the x-direction, which is also the stacking direction, with the electric field E polarized along the y-direction. To model the optical response of the SiC layer we consider a suitable Lorentzian fit to the experimental optical data, as reported in Ref. . So, we take the permittivity ɛ1 of the SiC layer to be:18], where propagation within the bulk material is prohibited. So, SiC has a phonon-polariton gap region that extends roughly from 10.3 to 12.6 μm (free space wavelength). We note that the form of permittivity dispersion given in Eq. (1) corresponds to EM waves varying in time as e−iωt [22, 23]. We take ɛ2=1 for the air slabs.
The absorptance, A, will be calculated with the Transfer Matrix Method (TMM) of Yeh, Yariv and Hong [24, 25], as A=1-R-T with R and T being the reflectance and transmittance respectively. We will see in the following, that to interpret and understand the absorptance behavior, the calculation and analysis of additional PC properties will be needed. We will also calculate the electric and magnetic field across the actual 1D PC of finite thickness with the TMM method. The band structure of the 1D PC, representing the dispersion relation for the propagating EM modes will be calculated with TMM by considering Bloch boundary conditions across the structural unit cell , thus modeling an infinitely extending 1D PC. Note, we will refer to the associated electric and magnetic field distributions for the infinite extending PC as modal fields.
3. Results and discussion on absorptance enhancement with the periodic SiC 1D PC
We consider the absorptance, A, through a 20 cell-thick PC for different filling ratios, f. We show the results versus free space wavelength, λfree, in Fig. 1(b) as red dashed lines for f=0.07, green dot-dashed lines for f=0.15 and blue dotted lines for f=0.30. The magenta vertical lines represent the spectral limits of the SiC phonon-polariton gap. The black solid line corresponds to the absorptance through a SiC block with thickness of 14.42 μm, which is equal to the total SiC thickness of the PC with f=0.07. Note, the latter represents a saturated absorptance value for bulk SiC in the phonon-polariton gap region, i.e. a thicker SiC slab won’t yield a higher absorptance.
For the cases of Fig. 1(b) we show the corresponding reflectance in Fig. 1(c). It is clear that reflectance is the major limiting factor of absorption. Our results suggest that parting a SiC block into thin slices can reduce the reflectance and in this manner enhance the absorption. We observe that the thinner these slices are the larger the absorption enhancement is, which also peaks more towards the center of the phonon-polariton gap. The higher filling ratio PC (f=0.30), is very little different from the bulk SiC structure. In other words, lower filling ratio polaritonic PCs are better absorbers in the frequency regime of the phonon-polariton gap. Our findings are consistent with observations on visible spectrum absorption with a plasmonic nanowire PC in the work of Ref. . Accordingly, in the following we focus only in the lower filling ratio structure of f=0.07.
The work of S. Y. Lin et al.  suggests that in PCs with lossy material constituents, absorption would be expected to peak around the PC band edge, were propagating modes are ultra-slow allowing for a larger interaction time between light and matter. We show the band structure for the SiC PC in Fig. 2(a). We plot the free space wavelength, λfree, with the photonic crystal (Bloch) wavevector q in dimensionless units [real part is shown in left panel and imaginary part is shown in the right panel]. The PC band edge in polaritonic PCs is actually not as sharply defined as for the case of lossless PCs. We will identify as the band-gap region the regime where the real part of the wavevector is in the vicinity of the Brillouin zone boundary (i.e. close to π/a), with a corresponding a large imaginary part. Accordingly, the result in Fig. 2(a) suggest an allowed band within SiC the phonon-polariton gap up until 11.5 μm, which we designate with the yellow shaded region. Notice the band edge is in agreement with the onset of a large reflectivity as we see in Fig. 1(c). It is very interesting to note that, absorptance actually peaks at about 10.9 μm, which is around the middle of the allowed propagation band, well away from the PC band edge. The absorptance at the peak wavelength is about 60%, representing an enhancement factor around 15 with respect to to bulk SiC.
In the following, we attempt to understand better the factors underpinning the occurrence of the absorptance peak away from the band edge. In plasmonic-based nanostructures intensity enhancement within the metal has been quoted as a criterion to optimize asborptance [3,5,6,8]. Therefore, we calculate here also the spectral behavior of the normalized electric-field intensity within the SiC layers in the PC. In order to do that we consider a long –two-hundred-cell–PC structure that emulates a semi-infinite PC . We consider the averaged intensity within the first, second and third SiC layer (counting from the interface) and normalize it with the intensity I0 of the incoming EM wave. Thus, the normalized averaged intensity enhancement Ienha is given by:Eq. (2) is taken at the front SiC face of the first, second and third PC cell in each case. Ey(x) represents the electric field distribution that is calculated with TMM for the two-hundred-cell-thick PC system.
We plot the results in Fig. 2(b), with black, red and green solid line for the normalized intensity average within the first, second and third SiC layer respectively. Interestingly, throughout the allowed band the normalized intensity enhancement seems to peak toward the band edge in the first cell, while its response gets relatively flat at the third unit cell. The results in Fig. 2(b) imply that the absorptance peak does not coincide spectrally with the intensity enhancement peak. Note, the latter is less than one implying that throughout the allowed band the normalized averaged intensity in the SiC layer is less than the incident light intensity. The intensity in the air layers is about 3 times larger but with similar spectral response as in the SiC layers.
We proceed now to shed light into the relationship between the energy velocity of the propagating mode and absorption. Frequently, the group velocity is used to characterize the energy velocity of the propagating mode. However, the energy and group velocities are equal only in the case of lossless constituents [15, 26]. Actually, these can be quite different when constituents with anomalous dispersion are considered, such as polaritonic media . Also, the group velocity captures the average energy velocity within the unit cell [15, 28]. However, the energy-velocity definition allows for spatial resolution within the individual SiC and air layers , which is crucial to obtain a measure of the light-matter interaction time within the SiC layer. We will consider in the following the energy velocity averaged within the SiC layer of an infinitely extending PC structure. This will be given by [27, 29]:Eq. (1). Note, that the proper form for the EM energy density is used here for the polaritonic SiC medium  that is consistent with Loudon .
The energy velocity result from Eq. (3) is plotted versus free space wavelength, λfree, as a solid red line in Fig. 2(c). The result is shown in units of the speed of light, c. We see that the energy velocity drops to almost zero from the band edge at about 11.5 μm, – designated with the blue solid vertical line in the figure. However, as also discussed above the absorptance peak, – designated with the blue dotted vertical line– is well away spectrally from this point. This result is contrary to the observations in Ref. , where absorptance maxima and energy velocity minima overlap spectrally at the PC band edge. We observe that in our polaritonic PC system the PC band edge ceases to be sharply defined . As a result, the energy velocity does not also sharply reduce to zero, but there is rather an extended region with a low ve. For the periodic PC, absorptance is peaking within the latter frequency regime but away from the point where ve almost drops to zero. We stress that the propagation mode at the absorptance peak wavelength is much faster in the SiC layer of the SiC-air PC, by at least an order of magnitude, in comparison to the one in bulk SiC. The energy velocity of the latter is shown with the green dashed line in Fig. 2(c).
We argue that the role played by the energy velocity in absorption is bi-fold. In particular, we assert that while slow light is conducive to absorption, due to a longer light-matter interaction time, at the same time it is an adverse factor causing an elevated reflectance. Accordingly, a slower mode with a larger interaction time does not necessarily imply also a greater absorption enhancement. We investigate further this aspect in the following section.
4. Energy velocity and reflectance
We attempt in this paragraph to develop a relationship between the energy velocity of light inside the SiC layers of the 1D PC and reflectance on the air-PC interface. We will consider again the semi-infinite PC medium. Note that, while in a PC with lossless constituents the energy velocity does not vary spatially within the individual constituent layers  this is not the case for lossy constituents. Inside the lossy SiC layers the energy velocity, ve has a spatial variation. We depict the spatial variation of ve in Fig. 3(a) within the first SiC PC layer, of the effectively semi-infinite PC for a free space wavelength of 10.9 μm. This represents a wavelength falling close to the absorptance peak in Fig. 1(a) for the twenty-cell PC [Note as we approach the semi-infinite limit the absorptance peak shown in Fig. 1(a) is inhomogeneously broadened]. We find that this spatial variation of ve within SiC remains exactly the same for layers inside the semi-infinite PC. Also we find that throughout the allowed PC band region [shaded region in Fig. 2(a)], the ve within the SiC layers retains this characteristic increasing-decreasing ve profile shape.
In the following we attempt to get a relation between the energy velocity, ve, at the interface of the semi-infinite PC and reflectance, R. Let α and β represent the electric field contributions just after the air-PC interface, propagating along the +x and −x directions respectively [where the axis system has been defined in Fig. 1(a)]. It can be easily shown with TMM that:Eq. (3) without the spatial average], after some math manipulations we obtain for the energy velocity at the air-PC interface: Eq. (3).
If a reflectionless mode couples into the PC, then effectively all energy can get absorbed for a sufficiently thick structure. Such mode would need to be slow at the interface, as implied by Eq. (8) (ɛfac ranges between roughly 50 and 175 within the allowed PC band regime). However, a comparison of ve, int from Eq. (8) and the energy velocity in the bulk SiC  yields that the PC medium would need to actually speed-up the light propagation inside the SiC material, in order to absorb the incoming light efficiently. We found that depending on the free space wavelength within the SiC polariton-gap regime, this speed-up factor ranges between 5 and 15.
Since we observe a rather smooth spatial variation of ve within the SiC layer, it is a fair assumption that the value of ve at the interface is close to the value averaged within the entire layer. Then, based on the expectation for the energy velocity of the reflectionless PC, and the spectral response of ve shown in Fig. 2(c), one would predict a near-perfect absorption around 11.35 μm. We however have not observed such near-perfect absorption for the periodic PC. This leads us to understand that there may be also a correlation between the type of energy velocity spatial profile at the interface (growing or decaying) and the reflectance R. We thus explore the existence of such correlation. The electric field expressions from the TMM method within the SiC interface layer, after some extensive math manipulations, yield the following expression for the energy velocity slope at the interface:Eq. (9) at the reflectionless condition, we obtain the following value for the energy velocity slope at the PC interface: Equation (10) suggests that for the reflectionless PC light should gradually slow down when entering the SiC-air PC. We note that this observation is not specific to SiC, but is general and applies to any air-layered-structure interface, comprised of any absorbing material including metals.
However, as we observe in Fig. 3(a) light initially speeds up and then slows down while coupling into the SiC-air PC. As we find this increasing-decreasing spatial profile for ve throughout the allowed PC-band region, this seems to pose a substantial limiting factor to the reflection minimization and accordingly to the optimization of absorptance. We therefore subsequently focus on investigating the possibility to manipulate the spatial ve profile at the PC-structure entry. For this reason we explore a PC design, where the first face is truncated. We note, that truncation of end faces has been considered before in lossless PCs as a route of transmission control [31, 32]. Our TMM based calculations for ve in the semi-infinite PC shows that the truncation leaves the energy velocity intact for layers well inside the PC medium.
Then, at the interface SiC layer, the ve spatial profile follows exactly the same functional form as in the overlapping periodic PC. As an example we show such ve profile for a truncated PC at 11.4 μm free space wavelength in Fig. 3(b) for a PC structure with a total of 20 cells. The SiC end face has half the width of a SiC layer within the bulk PC structure. We clearly observe a decaying ve profile from the interface, shown as a red dashed line in the figure. For comparison, the corresponding ve profile for the periodic semi-infinite PC is also shown as a black solid line. Note that in the figures we have taken x =0 at the front face of the second cell, that is the same for both periodic and truncated PC. The latter result essentially affirms that truncating the PC end face provides a route to manipulate at the interface. Equation (9) implies that this translates into manipulating the reflectance, R, and thus the absorptance of the PC structure. We explore such truncation based optimization route in the section that follows.
5. Near-perfect mid-IR absorption in the truncated PC design
We truncate the first layer to a thickness dint, while the remaining PC remains intact, as we depict in Fig. 4(a). We plot the absorptance, A, versus the free space wavelength, λfree, and the termination ratio tratio = dint/d1 in Fig. 4(b). The latter represents the thickness ratio of the truncated SiC layer and the SiC layers in the remaining periodic PC. The results suggest that the optimum absorptance occurs at a termination ratio close to 0.5 yielding essentially a near-perfect absorption. We depict the absorptance versus free space wavelength for the latter case separately in Fig. 4(c), where we can see clearly the near-perfect absorption at λfree ∼ 11.4 μm.
This results represents essentially an absorptance enhancement factor of 30, i.e. an increase of ∼ 2900%, in reference to the bulk SiC structure. For the peak wavelength we show also in the inset the absorptance versus the total number of photonic crystal cells. We see that with just three cells, corresponding to a total SiC thickness of ∼ 0.15λfree, we can obtain absorptance exceeding 50%. Thus, our paradigm structure achieves essentially the same IR absorption as a metallic PC , with a photon-matter interaction path that is more than an order of magnitude smaller. This is because the effective interaction path in metallic PCs is much smaller than the material thickness that is crossed by the EM wave, due to the ultra-small metallic skin depth at IR frequencies.
Thus, the truncation of the PC end face introduces a propagation mode where light gradually slows down as it enters which seems to facilitate the efficient in-coupling of slow light of ve, int ∼ 0.0115c, in this manner optimizing absorptance. We note, that while the absorptance peak red-shifts towards the band edge for the truncated PC, it still remains spectrally away from it.
Also, still the absorptance peak does not coincide spectrally with the intensity enhancement maximum, as we show in Fig. 5. The calculation was again performed with TMM for the semi-infinite truncated PC with tratio = 0.5. The averaged normalized intensity, Ienha is shown as black, red and green solid lines for the first, second and third SiC layer respectively. For comparison we show also in Fig. 5 the corresponding result for the first SiC layer of the periodic PC as the dashed blue lines. Evidently, the truncation does not have any impact on the spectral response of the intensity enhancement. The actual value of the latter increases across spectrum with the truncation, as more light couples into the structure. This is true for both SiC and air layers. This affirms our previous assertion that the truncation facilitates the in-coupling and thus optimizes the absorptance by manipulating the spatial profile of the light’s energy velocity at the entry.
It is very interesting to explore the angular and polarization dependency of the near-perfect absorptive property of the truncated SiC PC. We do so in the following section.
6. Angular and polarization response of the truncated PC design: Achieving polarization insensitive and polarization selective efficient absorptance
At normal incidence, TE and TH polarization are degenerate. However, for off-normal incidence these decouple. We depict TE and TH polarization incidence at the truncated PC in Figs. 6(a) and 6(b) respectively. We calculate then with the TMM for a truncated PC with tratio = 0.5 and thickness of 20 cells, the absorptance versus the angle of incidence θI, and free space wavelength, λfree, and we show the results in Figs. 7(a) and 7(b) respectively. We note that at each incident angle and frequency the truncation with tratio = 0.5 may not be always the optimum one. However, we find that overall that the optimum termination is close to a termination ratio of tratio = 0.5 across the spectrum of incident angles and frequencies within the phonon-polariton gap regime. Indicatively, we show for comparison the respective absorptance versus incident angle and free space wavelength for the fully periodic structure (tratio = 1.0), and a truncated PC with tratio = 0.25. The respective TE-polarization cases are shown in Figs. 7(c) and 7(e) and the respective TH-polarization cases in Figs. 7(d) and 7(f).
Indeed, looking closely we observe a near-perfect absorptance for the truncated PC design with tratio = 0.5 for different frequency-angle regimes. This implies that by tuning the angle of incidence we can manipulative the absorptive response of the PC structure. To see this more clearly we highlight this behavior by showing the absorptance versus free space wavelength for two particular incident angles: 30 deg. shown in Fig. 8(a) and 65 deg. shown in Fig. 8(b). At normal incidence, the two polarizations are degenerate, which means that we would get a near-perfect absorptance for both polarizations. The results in Fig. 8(a) suggest that this polarization insensitivity of the absorptance seems to survive even for a larger incident angle, but blue-shifts spectrally. The yellow shaded region in Fig. 8(a) represents a spectral regime where we can obtain more than 80% absorptance for both polarizations.
Now for an incident angle of 65 deg., the PC band-gap frequency regimes do not spectrally overlap for the TE and TH polarization case. We can confirm this by calculating the infinite-PC energy velocity along the propagation direction (x) versus the free space wavelength, for cases having a wavevector component along the z-direction [defined in Fig. 1(a)] that corresponds to an angle of incidence of 30 deg. and 65 deg. The results for the former case are shown in Fig. 9(a) while the results for the latter in Fig. 9(b) with a solid line for TE-polarization and a red dashed line for TH-polarization. As discussed in Sec. 4, the band-gap region can be identified by regions where ve is ultra-slow approaching zero. Thus, we observe in Fig. 9(a) that the band-edge is close for both TE and TH polarizations. This made it possible to get an efficient absorptance at about 10.9 μm, which is in the vicinity but away from the band-edge for both polarizations, as we saw in Fig. 8(a). On the other hand, this is not true for the case that corresponds to a 65 deg. incidence. We observe in Fig. 9(b) a spectral region until a free space wavelength of 11 μm where we have allowed TE-modes but forbidden TH modes. Conversely, within the spectral region of ∼ 11.5–12 μm the results of Fig. 9(b) suggest the existence of forbidden TE-modes but allowed TH modes.
These angle- and polarization-dependent allowed-band regions translate into an angle- and polarization-dependent absorption efficiency. We see clearly in Fig. 8(b) in the regime around 10.9 μm (cyan shaded region) an absorptance of more than 95% for TE-polarization and less than 10% for TH-polarization. Conversely, we observe around 12 μm (salmon shaded region) an absorptance of more than 95% for TH-polarization and less than 10% for TE-polarization. The later result demonstrates that our proposed truncated SiC PC can act as a polarization selective absorber at an angle of incidence of 65 deg.
In conclusion, we report here for the first time a non-metallic route to mid-IR near-perfect absorption with a SiC one-dimensional photonic crystal around the frequency of the bulk SiC phonon-polariton mid-gap. We have achieved the near-perfect absorption by considering ultra-thin SiC slices, –that allow for reduced reflectivity–, and in addition by tailoring the width of the input interface. Our careful investigation reveals that a slow light propagation simultaneously facilitates and impedes absorption, by allowing a larger light-matter interaction time and inducing a high reflection respectively. We find that efficient absorption occurs when the coupled mode is slow, but yet at least an order of magnitude faster than in the bulk SiC material. Also, we find that the absorptance peak can be spectrally well away from the PC band edge contrary to previous reports in metallic PCs .
Moreover, we demonstrate that a proper interface truncation manipulates the spatial distribution of the energy velocity at the interface. Our results show that a slow mode with decreasing energy velocity in the interface facilitates an efficient in-coupling and thus induces near-perfect absorption. We further show that by controlling the angle of incidence our proposed structure can act both as a polarization insensitive and as a polarization selective absorber. Our truncated PC paradigm opens a new avenue for absorption engineering within the phonon-polariton gap region of semiconductor and ionic crystal materials. We believe our study will inspire new highly efficient non-metallic absorber designs pertinent to mid-IR detection devices.
Financial support for the Ph.D. studentship of G.C.R.D. by the College of Engineering, Mathematics and Physical Sciences (CEMPS)-U. of Exeter is acknowledged.
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